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Spontaneous transitions between non-equilibrium patterns are characterised byhydrodynamic calculations of ideal straight roll steady state heat convection.The calculations are tested quantitatively against existing experimental data.It is shown that at a given Rayleigh number the final wave number depends uponwhether the initial state was the conducting state, or, in the case of thecross roll transition, the wave number of the initial straight roll state. Thefinal wave number does not correspond to the maximum or to the minimum heatflux (entropy production or dissipation), nor to the maximum sub-systementropy. In all cases the entropy of the total system increases monotonicallyduring the spontaneous transition. It is concluded that there does not existany single time thermodynamic property or variational principle fornon-equilibrium systems. It is further concluded that the second entropy is thetwo time variational principle that determines the optimum non-equilibriumstate or pattern.

Optimising Principle for Non-Equilibrium Phase Transitions and Pattern Formation with Results for Heat Convection